Abstract: New algorithms for automatically generating two-dimensional uniform and graded finite element meshes incorporate novel operators for the generation of nodes on an object's boundaries as well as interior. The algorithms begin with an initial triangulation of the object. The generation of additional nodes is used to generate a triangular mesh with the desired density. Most importantly, the algorithms guarantee a lower bound on the smallest angle in the whole mesh. This lower bound is 30.degree.. With this guaranteed lower bound, smoothing of the mesh or human intervention to improve the shape of the mesh becomes unnecessary.